How far can an animal fall and survive?

The topic is summarised nicely by J.B.S. Haldane in his essay
On Being the Right Size

You can drop a mouse
down a thousand-yard mine shaft; and, on arriving at the bottom,
it gets a slight shock and walks away, provided that the ground is
fairly soft. A rat is killed, a man is broken, a
horse splashes.

This assertion
depends on air resistance.
Drag acts against the force of gravity
on a falling object; the object reaches terminal
velocity when the force of gravity is equal to
the drag. Obviously the smaller the terminal velocity,
the better the chance of a happy landing. Very roughly
the terminal velocity is proportional to the size of the falling object.
So the terminal velocity of
a mouse is approximately 1% of that of an elephant.
To give specific examples, a parachutist
falls gently to earth at around 5 m/s
whereas a bacterium is travelling at
2.8 x 10-4 m/s.

In a study done by some New York city vets
(where apparently it is quite common for cats
to be defenestrated) it was found that
it takes a cat about 7 floors to hit
terminal velocity and that a cat can often survive this
with nothing worse than a nosebleed.

Beyond a simple matter of terminal velocity, survival is also a matter of the maximum speed at which a creature can survive a stiff impact... and that also varies with the size of a creature! An object's strength is proportional to the length scalesquared, but the amount of impulse required to stop the object is proportional to the mass, which is proportional to the length scale cubed. The larger the object, the more a collision with a sturdy object at a given velocity damages it.

One can also consider the xenobiological implications of this property of an animal. If, on some planet, the atmospheric pressure/gravity ratio is high (one would find this on large planets, particularly those with low density) then larger animals will be able to survive terminal velocity collisions. This could either make the creatures grow bigger or allow more of them adapt to being able to fall an essentially unlimited distance... or both, for different creatures in a given ecosystem.

(In response to a defunct writeup) I know this is more specific than what you were talking about, but a physically healthy adult cat is capable of flipping itself over to land feet downward, often before it has fallen one meter. My thesis advisor actually carried this experiment out, dropping his Tom onto cushions sufficiently soft that a head bonk wouldn't have harmed it. I only wonder how he managed to convince it to let him pick it up the second time.

Not only is it possible for a cat to survive a fall of 14 stories (or more), a cat is more likely to survive such a fall than one from only four stories (at least according to my freshman physics textbook; like everything else I learned during my freshman year, I trust it to be 100% correct). The reasoning is thus:

A cat has little ability to sense its absolute speed (which makes sense, since any good relativist will tell you that there's no such thing as an "absolute speed"). What a cat does have is a sense of proprioception, which tells the cat how fast it's accelerating. When a cat falls out of a window it begins accelerating at something close to 9.8 m/s; this really freaks the cat out, so it assumes its "I'm in danger" posture. Specifically, it tucks its head in, pulls its feet under its body, and curves its spine. This is a good thing for the cat to do if something has just thrown it; the important parts are towards the inside where they are less likely to get injured upon impact. Unfortunately, it also makes the cat much more aerodynamic.

Curled up like this, a cat cuts through the air suffering very little drag. The low surface area of this shape means the cat has a high terminal velocity (maybe 15m/s?). If it strikes the ground going this fast, it is unlikely to survive (at least not without serious injury).

Luckily, after the cat has fallen seven or eight stories, it begins to closely approach its terminal velocity. This means that the drag due to air resistance is equal to the force of gravity, so the cat is no longer accelerating. This allows the cat to calm down enough to more thoroughly appraise the situation. Cats apparently all have an instinctive understanding of certain laws of physics, or at least they know they're more likely to survive a fall it they spread themselves out. They stretch their legs and neck out, increasing their surface area, which decreases their terminal velocity. This slows the cat down considerably (and makes it look kinda like a flying squirrel). At this slower speed it can easily survive the fall (theoretically from any height, though 32 stories is the highest on record).